Bought a nice FN FAL a couple of months ago, but only recently had a chance to fire it...

I fired it a few times and then handed it off to my mate to shoot some, it doubled on her... Being that she is of small stature, I chalked it up to a bump fire and a wobbly folding table... I fired it from the same table and after a few rounds it doubled on me...

This I know, there was a definite click as the trigger reset and the gas adjustment was probably too high. I had never gotten around to adjusting it.

When I got home it dry fired, held the trigger back as I recycled the bolt... Taking very little pressure off of the trigger made it reset, and was able to fire with little effort.

Is this normal in the FAL? I'm thinking I need to hold the trigger back more firmly.

Doubling is sometimes caused by having the trigger springs reverses with the extractor spring.
The whole mechanism is a balancing act between spring tension & the sliding sear, if you have the similar-looking, but stronger spring installed this frequently causes doubling.
Some springs are color coded green to help prevent this.

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Allan Quatermain: “Automatic rifles. Who in God's name has automatic rifles”?

I have observed that extremely careful trigger control,very gradual increase in pressure,can find a way for the disconnect to sear handoff to fumble.Find the bump fire spot.Of course,it all should work flawlessly.A little more deliberate"fire" trigger pull can be better for semi-autos.

Is the rifle new from factory or it has been worked on? A trigger job perhaps?

I would check the sear engagement to make sure it is positive. Take out the trigger group and feel hammer movement while pulling on the trigger. The hammer should cam back before it falls. If the hammer moves forward you have negative engagement. It is prone to double or multiple, and it will need to be corrected.

Installed correctly.... Has FSE hammer trigger and sear set .... I've read a few places that these may double....

While the trigger works, in essence as a trigger, I get the heebie jeebies from it... Much unloaded experimenting, reveals that it's really easy to drop the hammer if you twitch ever so slightly on the return stroke

That's your issue right there.
A lot of FAL owners tried "lightening the pull" by changing springs & regrinding sear & hammer notches. Then they discovered why a battle rifle had a 5 Lb trigger pull in the first place.
The original design had a "safety sear" to prevent hammer follow & OOB discharge, but BATFE decided it was the "machinegiun part" & decreed it became title 3 with one cut in the receiver so they all have them removed in the U.S. unless they're a machine gun & registered as such.

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Allan Quatermain: “Automatic rifles. Who in God's name has automatic rifles”?

The FSE fire control group is your problem. I had a Century gun that worked fine but the trigger was less than good. I installed FSE components and there was a big improvement. Then it began to double. I contacted FSE and they were great. Sent out a new set right away.

I shot the gun a number of times over the years and a couple of years ago it began doubling again. I reinstalled the old Century parts and its been fine ever since.

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